Find the t* multiplier for a 99% confidence interval when the sample size is 45. Give your answer to 4 decimal places.

To find the \( t^* \) multiplier for a 99% confidence interval when the sample size is 45, we follow these steps:

1. **Determine the degrees of freedom (df):**
   The degrees of freedom for the \( t \)-distribution is given by \( df = n - 1 \), where \( n \) is the sample size.
   \[
   df = 45 - 1 = 44
   \]

2. **Locate the 99% confidence level in the \( t \)-distribution table:**
   For a 99% confidence interval, the level of significance \( \alpha \) is 0.01. Since the confidence interval is two-tailed, we split the significance level in half for each tail:
   \[
   \alpha/2 = 0.01/2 = 0.005
   \]

3. **Find the \( t \)-value corresponding to \( df = 44 \) and \( \alpha/2 = 0.005 \):**
   Using the \( t \)-distribution table or a calculator that provides \( t \)-values, we look up the \( t \)-value for \( df = 44 \) and \( \alpha/2 = 0.005 \).

Using a standard \( t \)-table or a statistical calculator, we find:

\[
t^* \approx 2.6955
\]

Thus, the \( t^* \) multiplier for a 99% confidence interval when the sample size is 45 is approximately \( 2.6955 \).

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. How is the degrees of freedom calculated in a \( t \)-distribution?
2. What is the difference between a \( t \)-distribution and a normal distribution?
3. How does the sample size affect the \( t \)-distribution?
4. What is the significance level \( \alpha \) in hypothesis testing?
5. How do you interpret a 99% confidence interval in practical terms?
6. How would the \( t^* \) value change if the sample size were increased to 100?
7. Why is the \( t \)-distribution used instead of the normal distribution for small sample sizes?
8. How do you use the \( t \)-distribution table to find critical values?

**Tip:** When using the \( t \)-distribution, always make sure to check the degrees of freedom and the corresponding \( \alpha \) value to find the correct \( t \)-multiplier.

Learn how to find the t* multiplier for a 99% confidence interval with a sample size of 45. Detailed step-by-step solution includes calculation of degrees of freedom, determining the significance level, and looking up the t-value in a t-distribution table. Suitable for undergraduate students and anyone interested in statistics and confidence intervals.

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